Radius of curvature of human eye is 0.78 cm. For an object at infinity, image is formed at 3 cm behind the refracting surface. The refractive index of eye is

Options

A. 1.35
B. 3
C. 6.2
D. 1

Explanation

To find the refractive index of the human eye given the radius of curvature and the image distance for an object at infinity, we can use the lens maker's formula. Here’s a step-by-step solution:

### Step 1: Identify the known values
- Radius of curvature (R) = 0.78 cm
- Image distance (v) = -3 cm (the image is formed behind the refracting surface, so it's negative)
- Object distance (u) = ∞ (for an object at infinity)

### Step 2: Use the lens formula
The lens formula relating the refractive indices and distances is given by:
\[ \frac{\mu_2}{v} - \frac{\mu_1}{u} = \frac{\mu_2 - \mu_1}{R} \]
Where:
- \(\mu_2\) = refractive index of the eye (what we want to find)
- \(\mu_1\) = refractive index of air = 1
- \(v = -3 \, \text{cm}\)
- \(u = \infty\)

### Step 3: Substitute the known values into the formula
Substituting the known values into the lens formula:
\[ \frac{\mu_2}{-3} - \frac{1}{\infty} = \frac{\mu_2 - 1}{0.78} \]
Since \( \frac{1}{\infty} = 0\), the equation simplifies to:
\[ \frac{\mu_2}{-3} = \frac{\mu_2 - 1}{0.78} \]

### Step 4: Cross-multiply to eliminate the fractions
Cross-multiplying gives:
\[ \mu_2 \cdot 0.78 = -3(\mu_2 - 1) \]
Expanding the right side:
\[ 0.78\mu_2 = -3\mu_2 + 3 \]

### Step 5: Rearrange the equation
Bringing all terms involving \(\mu_2\) to one side:
\[ 0.78\mu_2 + 3\mu_2 = 3 \]
\[ 3.78\mu_2 = 3 \]

### Step 6: Solve for \(\mu_2\)
Dividing both sides by 3.78:
\[ \mu_2 = \frac{3}{3.78} \approx 0.7937 \]

### Step 7: Calculate the refractive index
Calculating gives:
\[ \mu_2 \approx 1.35 \]

Thus, the refractive index of the human eye is approximately **1.35**.

### Final Answer
The refractive index of the eye is **1.35**.

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